Integral Calculus -7
20 Questions MCQ Test | Integral Calculus -7
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The length of the cycloid with parametric equation x(t) = (t + sin t), y(t) = (1 - cos t) between (0, 0) and (π, 2) is:
The given curve is
x(t) = (t + sin t), y(t) = (1 - cos t)
Which of the following is/are true?
1. 
2. 
3. 
4. If f(x) ≥ φ(x) and both functions are integrable in [a, b]
1.
2.
3.
4. If f(x) ≥ φ(x) and both functions are integrable in [a, b]
is revolved about major and minor axis respectively. Then the ratio of these solids made by two revolutions is:
Case (i): When ellipse is rotated about major axis:
Take a small disc at a length of x from the centre of thickness dx. Then the volume of solid obtained by rotation will be ∫(−a to a)(Area)dx
Area of disc =πr2
r can be calculated from the equation of ellipse as
x2/a2 + r2/b2 = 1
⇒ r2 = b2(1−x2/a2) = b2/a2(a2−x2)
∴Volume of major axis = ∫(−a to a)πa2/b2(a2−x2)dx
= [πb2x − πb2x3/3a2](−a to a)
= 4/3πab2
Case (ii): When ellipse is rotated about minor axis:
Following similar procedure as case (i),
r2 = a2/b2(b2 − y2)
In this case, the area will be integrated w.r.t dy as it is rotated about the Y-axis.
∴Volume of minor axis = ∫(−a to a)πa2/b2(b2−y2)dx
= [πa2y − πa2y3/3b2](−a to a)
= 4/3πab2
∴ Volume about major axis/Volume about minor axis
= b/a
...(i)
...(ii)
The surface area of the segment of a sphere of radius a and height h is given by:
Let the sphere be generated by the revolution about the x - axis of the circle
x2 + y2=a2 ...(i)
Let OA =a, OC = b and OB = b + h
Hence The required surface
then the value of a is:
The moment of inertia about the axis of y of the region in the xy-plane bounded by y = 4 - x2 and the x-axis, is, proportional to
The given curve is y = 4x2
Hence Required moment of inertia
...(i)
...(ii)
Given that
y = loge
The intrinsic equations of the cardioids r = a (1 - cos θ) and r = a(1 + cos θ) measured from the pole are:
The equations of given cardioids are
r= a (l -cos θ) ...(i)
r= a (1 + cos θ) ...(ii)
Intrinsic equation for cardioid (i):
...(iii)
Consider the Assertion (A) and Reason (R) given below:
Assertion (A)
Reason (R) - sin x is continuous in any closed interval [ 0 ,t] .
The correct answer is:
Let
The perimeters of the cardioids r = a (1 - cos θ) and r = a (1 + cos θ) differ by:
The curves are
r = a(1 - cos θ) ...(i)
r = a(1 + cos θ) ...(ii)
The perimeter for curve (i) is
Hence s1 - s2 = 8a - 8a = 0.
The length of the arc of the curve 6xy = x4 + 3 from x = 1 to x = 2 is:
The given curve is
6xy = x4 + 3
Thus Length of the arc between x
= 1 and x = 2 is:
The area of the region in the first quadrant bounded by the y-axis and curves y = sin x and y = cos x is:
The given curves are
y = sin x ...(i)
y = cos x ...(ii)
The curves intersect at a point B 
Hence Required area = Area OABO
The line y = x + 1 is revolved about x-axis. The volume of solid of revolution formed by revolving the area covered by the given curve, x-axis and lines x = 0, x = 2 is:
The given curve is y = x + 1. This represents a straight line
Hence The required volume
The length of the complete cycloid x = a (θ + sin θ), y = a(1 - cos θ) is given by:
The equations of the curve are
x = a(θ + sin θ), y = a(1 - cos θ)
Hence The entire length of the curve
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